Mathfolis

Blocking & Matched Pairs Designs

Unit 3 · Collecting Data

What AP Stats asks here

Blocking groups similar subjects together to remove a known source of nuisance variation. Matched pairs is the special case of two-subject blocks — twins, paired plots, or before/after on the same person. The recurring AP trap: a student treats before-and-after measurements as two independent samples and runs the wrong test.

Designs at a glance

Completely randomized
no obvious nuisance variable\text{no obvious nuisance variable}
Randomized block
group, then randomize within blocks\text{group, then randomize within blocks}
Matched pairs
2-subject blocks (twins or within-subject)\text{2-subject blocks (twins or within-subject)}

Analysis matches design

Independent groups
two-sample t-test\text{two-sample } t\text{-test}
Matched pairs
paired t-test on the differences\text{paired } t\text{-test on the differences}
AP Tip: Block on any variable that explains a lot of variation in the response — soil region, sex, prior ability. The blocking variable should correlate with the response, not with the treatment.
Caution: Before/after measurements on the same subject are NOT two independent samples. Treating them that way inflates the standard error and loses statistical power.
Type 1

When to block

Block when a known variable explains a meaningful share of the response variation. Stratifying samples is to descriptive work what blocking experiments is to causal work.

Example 1
An experiment tests two fertilizers on tomato yield across a field with varying soil quality. Should the design block on soil quality, and why? (A) No — randomization is enough; blocking is unnecessary. (B) Yes — block by soil region (e.g., high vs low quality) and randomize fertilizers within each region, removing soil quality from the comparison. (C) Yes — but block by the day of the week. (D) No — instead increase the sample size by 10x.

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Type 2

Matched pairs design

Matched pairs is the design when each subject is paired with a strongly-similar partner (twin) or with themselves (before/after, crossover). The unit of analysis is the difference.

Example 2
Researchers compare a new drug to a placebo. Each of 40 volunteers takes both, in random order with a washout period in between. Drug response is compared to placebo response per volunteer. Which design is this? (A) Two independent samples — 40 on drug and 40 on placebo. (B) Matched pairs (crossover) — each volunteer is their own control. (C) Stratified sampling. (D) Cluster sampling.

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Type 3

Analysis that matches the design

Paired data require a paired analysis. A two-sample test on paired data inflates variance because it ignores the strong within-pair similarity.

Example 3
A student measured blood pressure before and after exercise on 25 subjects, then used a two-sample tt-test treating 'before' and 'after' as two groups. Which analysis is correct? (A) The two-sample tt-test is fine because the two groups have the same n. (B) Compute the difference (after − before) for each subject and run a one-sample (paired) tt-test on the 25 differences. (C) Use a chi-square test of independence. (D) No test is needed; just report both means.

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Blocking & Matched Pairs Designs | AP Statistics — Mathfolis